Spatial noise correlations in a Si/SiGe two-qubit device from Bell state coherences

We study spatial noise correlations in a Si/SiGe two-qubit device with integrated micromagnets. Our method relies on the concept of decoherence-free subspaces, whereby we measure the coherence time for two different Bell states, designed to be sensitive only to either correlated or anticorrelated noise, respectively. From these measurements we find weak correlations in low-frequency noise acting on the two qubits, while no correlations could be detected in high-frequency noise. We expect nuclear spin noise to have an uncorrelated nature. A theoretical model and numerical simulations give further insight into the additive effect of multiple independent (anti)correlated noise sources with an asymmetric effect on the two qubits as can result from charge noise. Such a scenario in combination with nuclear spins is plausible given the data and the known decoherence mechanisms. This work is highly relevant for the design of optimized quantum error correction codes for spin qubits in quantum dot arrays, as well as for optimizing the design of future quantum dot arrays.

Figure 1

(a) Scanning electron micrograph of a similar Si/SiGe device as used in the measurements, showing the depletion gates used to define the potential landscape in the 2D electron gas accumulated by the yellow shaded gates (drawn digitally). Purple and orange circles indicate the estimated positions of the two dots, occupied by one electron each, and the ellipse indicates a sensing quantum dot. Two-qubit operations are controlled via gate voltage pulses applied to gates P1 and P2, and microwave signals for single-qubit control are applied to gates MW1 and MW2. The contours of cobalt micromagnets are indicated by the dashed black lines. (b) Energy level diagram for two qubits in an inhomogeneous magnetic field, giving rise to a difference in Zeeman energy between the two qubits.

Figure 2

$T_{2,|\mathrm{\Psi }\rangle }^{*}$ extracted from Eq. (1): (a) as a function of correlation factor $\rho$ and noise amplitude ${\sigma }_1={\sigma }_2$, and (b) as a function of ${\sigma }_1$ and ${\sigma }_2$ for $\rho =1$. Insets show the corresponding images for $T_{2,|\mathrm{\Phi }\rangle }^{*}$. Contours correspond to (0.5, 0.75, 1.0, 1.25, 1.5, 1.75) $\mu \mathrm{s}$. In all images an uncorrelated noise contribution corresponding to a Bell state coherence time of $2.0\mu \mathrm{s}$ is added to prevent singularities.

Figure 3

(a) and (c) Circuit diagrams for two-qubit experiments analogous to the measurement of Ramsey fringes. The gate sequences are designed such that single-qubit rotations are always applied simultaneously to both qubits, avoiding idle times that would lead to faster dephasing. Here $C{Z}_{ij}|m,n\rangle ={(-1)}^{\delta (i,m)\delta (j,n)}|m,n\rangle$ for $i,j,m,n\in \{0,1\}$ are the primitive two-qubit gates, constructed from a CZ gate with duration $t=\pi \hslash /J$ and single-qubit rotations [25]. (b) and (d) Typical $|00\rangle$ return probability as a function of delay time for (b) $|\mathrm{\Psi }\rangle$ and (d) $|\mathrm{\Phi }\rangle$. The data are fit with a sinusoidal function with Gaussian decay, $P_{|00\rangle }\propto e^{-{\left(t/T_2^{*}\right)}^2}$. Error bars are based on a Monte Carlo method by assuming a multinomial distribution for the measured two-spin probabilities and are $\pm 1\sigma$ from the mean [25]. We attribute the slight difference in oscillation frequency between (b) and (d) to crosstalk effects during frequency calibration, as for example observed in Refs. [23, 25, 41]. (e) Scatter plot of decay times for $|\mathrm{\Psi }\rangle$ and $|\mathrm{\Phi }\rangle$ for two measurement runs separated by $\sim 50\mathrm{h}$ (points and crosses). Every data point is averaged over $\sim 100\min$. The average coherence times are $513\pm 8$ and $387\pm 6\mathrm{ns}$ for $|\mathrm{\Psi }\rangle$ and $|\mathrm{\Phi }\rangle$, respectively. Error bars are $\pm 1\sigma$ from the mean.

Figure 4

Scatter plot of the two-qubit coherence times obtained in Hahn-echo style measurements for $|\mathrm{\Psi }\rangle$ and $|\mathrm{\Phi }\rangle$, from a fit to the data with an exponentially decaying sinusoidal function ($P_{|00\rangle }\propto e^{-t/T_2^{*}}$). Triangles represent data points where the Hahn echo pulses applied to both qubits are rotations around the $\widehat{x}$ axis. For the circles, the rotation of Q1 is around $\widehat{x}$ and the rotation of Q2 is around $\widehat{y}$. Data points are averaged over $\sim$[47, 66, 100, 148] min. The average two-qubit Hahn echo coherence times are $2.03\pm 0.09$ and $1.98\pm 0.09 \mu \mathrm{s}$ for $|\mathrm{\Psi }\rangle$ and $|\mathrm{\Phi }\rangle$, respectively. Error bars are $\pm 1\sigma$ from the mean.